Unbiased Estimation | Bessel's Correction

Bessel’s correction is an approach to remove bias when there is only sample data available to calculate the population variance.

Why is there a bias while calculating sample variance to estimate population variance?

Consider the following example:

Let’s calculate the population variance:

The result show as variance of ~6.5 for the entire population. In this case we have the entire population data available to calculate the population variance.

Let’s consider the case where the population data is not available and we need to estimate the population variance using the sample data available.

Take random sample from this population:

Calculating the variance for the random sample of [9, 8, 5] from the population set:

The sample variance results in a value of ~2.8 which is not representative of the population variance. There is a significant difference in the population variance of ~6.5 vs. the sample variance of ~2.8. This error is caused by limited sample data - the values in which are not the true representative of the population values.

Therefore, the estimation of the population variance is biased is biased due to lack of enough sample data, i.e., the more samples we will have the more likely we are able to end up with a value which is correct representative of the population values.

But how do we handle this bias in estimation when there is not enough sample data available?

This is done by Bessel’s correction that returns an unbiased estimation of the population variance using sample data by dividing the sum of square by (n-1) instead of n.

Calculating the sample variance with Bessel’s correction:

Dividing by (n-1) results in a sample variance which is much closer to population variance as compared to the sample variance obtained from dividing by n.